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IB Mathematics: Applications and Interpretation HL · 12.5: Functions · 12.5.6

Name Date

Stretches and reflections Problem set B

Sketch by moving the key points: vertex, intercepts. Outside the bracket changes y, inside changes x. No calculator needed.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The graph of \(y=f(x)\) is shown. It has its maximum at (1, 12), crosses the x-axis at x = −1 and x = 3, and crosses the y-axis at (0, 9). Sketch each graph on its own axes, marking the vertex and the intercepts.

    −3−2−112345−22468101214xyy = f(x)
    1. (a) \(y=3f(x)\)
    2. (b) \(y=\frac{1}{4}f(x)\)

  2. The graph of \(y=f(x)\) is shown. It has its minimum at (3, −4), crosses the x-axis at x = 1 and x = 5, and crosses the y-axis at (0, 5). Sketch each graph on its own axes, marking the vertex and the intercepts.

    −2246−6−4−2246xyy = f(x)
    1. (a) \(y=f(3x)\)
    2. (b) \(y=f(\frac{x}{2})\)

  3. Describe the transformation that maps the graph of \(y=4x^2-10x\) onto the graph with each equation.

    1. (a) \(y=10x^2-25x\)
    2. (b) \(y=2x^2-5x\)
    3. (c) \(y=36x^2-30x\)

  4. Given that \(f(x)=7x^2-14x\):

    1. (a) show that \(3f(x)=21x^2-42x\);
    2. (b) describe the transformation that maps \(y=f(x)\) onto \(y=21x^2-42x\).

  5. The graph of \(y=f(x)\) has a vertex at the given point. Find the vertex of \(y=-f(x)\).

    1. (a) \((−4, 3)\)
    2. (b) \((5, −8)\)

  6. The graph of \(y=f(x)\) has a vertex at the given point. Find the vertex of \(y=f(-x)\).

    1. (a) \((−8, 2)\)
    2. (b) \((7, −3)\)

  7. Describe each graph as a sequence of stretches and reflections of \(y=f(x)\).

    1. (a) \(y=-4f(x)\)
    2. (b) \(y=3f(4x)\)
    3. (c) \(y=-\frac{1}{3}f(x)\)
    4. (d) \(y=-2f(-3x)\)

In context

  1. The cross-section of a garden pond is modelled by \(y=f(x)\), where x and y are in metres and the water surface is the x-axis. The pond is 10 m wide and 2.5 m deep at its deepest point (5, −2.5). The graph is shown. A landscaper tests three other ponds.

    51015202530−5−4−3−2−112xyy = f(x)
    1. (a) Pond 1 is \(y=\frac{8}{5}f(x)\). What is the greatest depth, and where does the pond meet the water surface?
    2. (b) Pond 2 is \(y=f(\frac{x}{2})\). Find the greatest depth and the width of the pond at the surface.
    3. (c) Write down the equation of a pond with the same depth as the original but one third of the width, in the form \(y=f(qx)\), and give the value of \(q\).

  2. A survey of a river bed gives its profile as the graph of \(y=f(x)\), in metres, with the deepest point at (−5, −3). A model of the river is built as the mirror image \(y=f(-x)\), and a second model is the graph turned over, \(y=-f(x)\).

    1. (a) Find the deepest point of the mirror-image model.
    2. (b) Find the highest point of the turned-over model.

Stretch

  1. Describe the transformation that maps the graph of \(y=4x^2-6x\) onto the graph of \(y=16x^2+12x\).

  2. The point (8, 5) lies on the graph of \(y=f(x)\). The point (−4, 15) lies on the graph of \(y=pf(qx)\), where p and q are constants. Find p and q, and describe the transformation.

jmaths.xyz · AI HL · 12.5.6 Stretches and reflections